
==== Front
iScience
iScience
iScience
2589-0042
Elsevier

S2589-0042(24)01930-8
10.1016/j.isci.2024.110705
110705
Perspective
Pushing the limits of ultrafast diffraction: Imaging quantum coherences in isolated molecules
Tang Zilong 1
Jarupula Ramesh 1
Yong Haiwang hyong@ucsd.edu
12∗
1 Department of Chemistry and Biochemistry, University of California, San Diego, La Jolla, CA 92093, USA
2 Program in Materials Science and Engineering, University of California, San Diego, La Jolla, CA 92093, USA
∗ Corresponding author hyong@ucsd.edu
10 8 2024
20 9 2024
10 8 2024
27 9 110705© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Summary

Quantum coherence governs the outcome and efficiency of photochemical reactions and ultrafast molecular dynamics. Recent ultrafast gas-phase X-ray scattering and electron diffraction have enabled the observation of femtosecond nuclear dynamics driven by vibrational coherence. However, probing attosecond electron dynamics and coupled electron-nuclear dynamics remains challenging. This article discusses advances in ultrafast X-ray scattering and electron diffraction, highlighting their potential to resolve attosecond charge migration and vibronic coupling at conical intersections. Novel techniques, such as X-ray scattering with orbital angular momentum beams and combined X-ray and electron diffraction, promise to selectively probe coherence contributions and visualize charge migration in real-space. These emerging methods could further our understanding of coherence effects in chemical reactions.

Graphical abstract

Natural sciences; Physics; Optics

Subject areas

Natural sciences
Physics
Optics
==== Body
pmcIntroduction

Quantum coherence plays a crucial role in determining the outcome and efficiency of photochemical reactions and ultrafast dynamics.1,2 This quantum phenomenon manifests in molecules in three distinct yet interconnected forms.

Vibrational coherences

Arising from superpositions of vibrational states, these coherences involve energy gaps typically ranging from ten to a few hundred millielectronvolt (meV). This corresponds to nuclear motions on the femtosecond timescale.

Electronic coherences

Generated by superpositions of multiple electronic states with larger energy gaps compared to vibrational states, typically on the order of 1–10 eV. These result in rapid electron dynamics in molecules with coherence periods ranging from sub-femtoseconds to attoseconds.

Vibronic coherences

Resulting from the coupling of both electronic and vibrational degrees of freedom, involving energy gaps that span a wide range. One important example in molecular systems is conical intersections (CIs), where two or more adiabatic potential energy surfaces become degenerate, resulting in strong non-adiabatic couplings. These couplings play critical roles in virtually all reaction dynamics, such as light-induced isomerization, charge transfer, and energy transfer.

To fully understand the role of quantum coherence in chemical reactions, it is essential to resolve all these forms in real space and time as the reaction occurs. This requires a comprehensive understanding of both nuclear and electronic dynamics, particularly the coupled electronic and nuclear motions. Unraveling the interplay between these different manifestations of quantum coherence and their impact on chemical reactions has the potential to revolutionize our understanding of chemical processes and advance quantum computing technologies.3,4,5

The nuclear motion of molecules, specifically the role of vibrational coherences in chemical reactions, has been extensively investigated in recent years. Traditionally, molecular structures have been determined using X-ray crystallography and electron diffraction on crystalline samples. The development of time-resolved gas-phase X-ray and electron diffraction techniques has opened new avenues for investigating the structural dynamics of isolated molecules on picosecond timescales.6,7,8,9,10 By employing pump-probe schemes with ultrafast laser pulses, the real-time dynamics of chemical reactions, including transition states,11,12,13 can be captured with atomic spatial resolution. Recent advancements in ultrafast gas-phase X-ray scattering and mega-electron-volt ultrafast electron diffraction (MeV-UED) have further pushed the boundaries of temporal and spatial resolution,14,15,16,17,18 enabling the tracking of femtosecond atomic nuclear motion in molecules governed by vibrational coherence.

Despite recent progress in studying vibrational coherences in molecules, the direct observation of attosecond electron dynamics originating from purely electronic coherences and coupled electron-nuclear dynamics arising from vibronic coherences remains a challenge. Purely electronic coherences can lead to attosecond charge migration and energy transfer processes within the molecule, which play essential roles in photochemical reactions and light-harvesting systems. While recent ultrafast spectroscopic studies have begun to shed light on these attosecond electron dynamics,19,20,21,22,23,24 capturing these phenomena with diffraction methods has not been achieved, mainly due to the lack of required attosecond temporal resolutions. This is expected to become possible in the near future with ongoing efforts to develop attosecond X-ray free-electron lasers (XFELs) and attosecond electron pulses.25,26,27,28,29 Furthermore, experimental signatures of vibronic coherences are often overshadowed by stronger signals originating from electronic populations or vibrational excitations, necessitating novel experimental concepts and theoretical methods to distinguish these subtle coherent phenomena amidst complex molecular dynamics.

This perspective aims to provide a brief overview of the current state-of-the-art ultrafast gas-phase X-ray scattering and electron diffraction techniques, particularly their experimental advances in studying nuclear dynamics driven by vibrational coherence. By discussing recent advances and future opportunities in extending these techniques to investigate the other two forms of quantum coherences (i.e., electronic and vibronic coherences), we hope to inspire the development of new experimental and theoretical approaches that can provide a comprehensive understanding of all three forms of quantum coherences in photoinduced processes. Such advancements could lead to the ability to directly observe and control attosecond electron dynamics as well as coupled electron-nuclear dynamics at conical intersections. The question of how to control quantum coherences in molecules to influence reaction outcomes has led to the emergence of a new field called “attochemistry.”30,31 This emerging field has potential far-reaching applications in areas such as photocatalysis, light-harvesting, and quantum sensing.32,33,34

Theory of coherence signals in time-resolved molecular diffraction

Time-resolved diffraction techniques have emerged as powerful tools for probing the quantum dynamics of molecules with unprecedented spatial and temporal resolution. The theoretical description of time-resolved molecular diffraction has been extensively studied in the 1990s by Wilson, Cao, and co-workers.35,36,37 It has been revealed lately that in addition to the well-known elastic and inelastic scattering components, time-resolved diffraction signals contain distinct contributions related to electronic coherence.38,39,40,41,42 Such coherence signal, arising from the superposition of electronic (or vibronic) states, plays a crucial role in the ultrafast dynamics of molecules, such as charge migration and conical intersection passage. By probing the time evolution of the coherence signal, time-resolved diffraction techniques can provide new insights into the fundamental quantum processes that govern chemical reactivity. In this section, we present a brief overview of the time-resolved diffraction theory, with an emphasis on its connection to the quantum coherences in molecules.

The time-dependent molecular many-electron wavefunction prepared by a pump pulse may be expanded as(Equation 1) Ψ(r,R,t)=∑ici(t)χi(R,t)φi(r,R)

where i labels the adiabatic electronic states, χi(R,t) is the normalized nuclear wavepacket in the adiabatic electronic state ϕi(r,R), r and R are the electronic and nuclear coordinates, t is time and ci is the electronic state amplitude.

Using Equation 1, the time-evolving electronic charge density in real space is given byσtotE(r,t)=∑ijρji(t)〈χi(t)|σˆijE(r)|χj(t)〉

=∑iρii(t)〈χi(t)|σˆiiE(r)|χi(t)〉+2R[∑j>iρji(t)〈χi(t)|σˆijE(r)|χj(t)〉]

(Equation 2) =σpopE(r,t)+σcohE(r,t)

Here σˆE(r) is the electronic charge-density operator, ρ is the density matrix operator, ρii(t)=ci∗(t)ci(t) are real numbers representing the electronic populations at time t while the coherence terms, ρji(t)=ci∗(t)cj(t) with j≠i, consists of complex numbers. The vibronic coherence is obtained from the combined electronic-nuclear wavefunction as the overlap of the nuclear wave packets. The total electronic charge density contains contributions from both time-evolving electronic population density σpopE(r,t) and coherent density σcohE(r,t).

For the case of attosecond electron dynamics in molecules where the nuclei are static, the σtotE(r,t) in Equation 2 can be simplified as(Equation 3) σtotE(r,t)=σpopE(r)+σcohE(r,t)=∑iρiiσiiE(r)+∑i≠jρji(t)σijE(r)

We note that the electronic population term ρii is time-independent in the absence of non-adiabatic transitions between different electronic states. The time-dependence of the total charge density is thus contributed solely by the purely electronic coherence term σcohE(r,t) in Equation 3, which is responsible for the time-evolving electron dynamics in molecules.

The theoretical description of the time-resolved diffraction signal in this article is based on the off-resonant single-molecule (gas-phase) time-resolved X-ray/electron diffraction in the minimal coupling picture.40,43The time-resolved single-molecule diffraction signal is given by44(Equation 4) S(q,T)∝W0(Δω)∫dt|AX(t−T)|2S˜(q,t)

where AX(t−T) is the X-ray/electron probe pulse vector potential at delay time T from the pump pulse, q is the scattering momentum transfer, W0(Δω) is a window function for a frequency detection widow Δω and S˜(q,t) is the time-dependent molecular response in X-ray/electron diffraction.42,45,46 We assume a window function much broader than the relevant electronic transition energies of the system so that W0(Δω) is independent of the molecular response.

The time-resolved molecular response in X-ray diffraction, S˜(q,t), in Equation 4 is given by(Equation 5) S˜XRD(q,t)=∑ijkρji(t)〈χi(t)|σˆikE(−q)σˆkjE(q)|χj(t)〉

where σˆkjE(q) is the electronic charge-density operator in momentum-space. Similar to the Equation 2, the time-resolved X-ray diffraction signal can be partitioned into the sum of contributions from electronic populations S˜popXRD(q,t) and coherences S˜cohXRD(q,t),(Equation 6) S˜XRD(q,t)=S˜popXRD(q,t)+S˜cohXRD(q,t)

where(Equation 7) S˜popXRD(q,t)=∑iρii(t)∑k〈χi(t)|σˆikE(−q)σˆkiE(q)|χi(t)〉

(Equation 8) S˜cohXRD(q,t)=2R[∑j>iρji(t)∑k〈χi(t)|σˆikE(−q)σˆkjE(q)|χj(t)〉]

Unlike X-ray scattering, which is dominated by the molecular electronic charge density, electron scattering originates from the electrostatic Coulomb interaction of the incoming electrons with both molecular electrons and nuclei. The interaction between charged particles resulting a 1/q4 term in the Rutherford scattering.47 The time-resolved molecular response in Equation 4 becomes S˜(q,t)=1q4S˜UED(q,t) where S˜UED(q,t) is given byS˜UED(q,t)=∑ijkρji(t)〈χi(t)|σˆikE(−q)σˆkjE(q)|χj(t)〉+∑iρii(t)〈χi(t)|σˆiiN(−q)σˆiiN(q)|χj(t)〉+2R[∑ijρji(t)〈χi(t)|σˆijE(−q)σˆjjN(q)|χj(t)〉]

(Equation 9) =S˜elecUED(q,t)+S˜nuclUED(q,t)+S˜mixedUED(q,t)

Here σˆiiN(q) is the nuclear charge-density operator in momentum-space, S˜elecUED(q,t) is the electronic contribution to the signal which is identical to S˜XRD(q,t) given by Equation 5. In addition to the electronic contribution, S˜nuclUED(q,t) is the nuclear contribution to the signal and S˜mixedUED(q,t) is the mixed electronic-nuclear interference in electron diffraction which are distinct from X-ray diffraction as illustrated in Figure 1.Figure 1 Loop diagrams for single-molecule time-resolved X-ray diffraction (blue box) and electron diffraction (black box)

Recent advances in ultrafast gas-phase diffraction experiments

Ultrafast gas-phase diffraction experiments, including X-ray scattering and electron diffraction, have undergone significant advancements in recent years. These experiments typically employ a pump-probe scheme (see Figure 2), where a pump pulse excites the molecule to a higher electronic state, followed by a delayed probe pulse (either X-ray or electron beam) that interacts with the excited molecule. By varying the time delay between the pump and probe pulses, the structural evolution of the molecule can be followed in real-time.Figure 2 Experimental set-ups for ultrafast gas-phase X-ray (top) and electron diffraction (bottom)

The advent of XFELs, with their ultrashort pulse durations and extreme brightness,48,49 has revolutionized ultrafast X-ray scattering for studying structural dynamics in free molecules. Proof-of-principle femtosecond gas-phase X-ray scattering experiments conducted at Linac Coherent Light Source50,51 resulted in a "molecular movie" of the ring-opening reaction of 1,3-cyclohexadiene.52 Subsequent improvements, such as a newly designed diffractometer, shot-to-shot X-ray intensity fluctuation calibration, generalized detector geometry calibration, and various data processing advancements,53,54 have enhanced signal-to-noise ratios and enabled detailed studies of molecules and chemical dynamics in excited states. These studies include the investigation of ultrafast nuclear motions during chemical reactions,55,56,57,58 the determination of polyatomic molecular structures in electronically excited states,59 the observation of bond elongation and contraction during charge transfer,60 and the direct measurement of the redistribution of molecular electron density immediately after photoexcitation.61 Further examples include studies of chemical kinetics,62,63 anharmonicities, and correlations.64

Concurrently, gas-phase MeV-UED has emerged as another powerful tool for probing structural dynamics in isolated molecules. Pioneering MeV-UED studies at SLAC National Accelerator Laboratory demonstrated the capability of imaging vibrational wavepacket motion in I2 with sub-angstrom spatial and 230 fs time resolution, confirming MeV-UED’s potential for tracking atomic motion in gas-phase molecules.65 Since then, MeV-UED has been extended to study reaction dynamics in more complex molecules, such as imaging the nonadiabatic dynamics of CF3I,66 observing the isomerization of hexatriene following the photoinduced ring-opening of 1,3-cyclohexadiene,67 capturing the "pericyclic minimum" intermediate in α-terpinene,68 and revealing competing dissociation channels in CS2.69 Furthermore, MeV-UED has demonstrated the ability to simultaneously record both nuclear and electronic dynamics by exploiting inelastic scattering signals, as shown in studies of pyridine and ammonia molecules.70,71

Future perspectives: Tracking quantum coherences in molecules

Recent advancements in both X-ray scattering and electron diffraction have established ultrafast gas-phase scattering as powerful tools for observing structural dynamics originated from vibrational coherences on femtosecond timescales and sub-angstrom length scales. The ongoing development of attosecond X-ray free electron lasers and electron sources promises to open new frontiers in the study of ultrafast molecular dynamics. By pushing the temporal resolution to the attosecond regime, these techniques could enable the direct observation of attosecond electron dynamics and vibronic coupling in molecules.

Probing attosecond electron dynamics driven by electronic coherences

The interaction between a molecule and a broadband light source can lead to the creation of a superposition of electronic states, causing time-dependent variations in the molecule’s charge density. This phenomenon, referred to as charge migration, occurs on attosecond timescales and is solely a result of electronic coherence. Real-time experimental observation of charge migration is vital for optimizing and controlling fundamental events in various chemical and biophysical processes and has been a primary focus of attosecond molecular science.72,73,74 However, directly observing the time-evolving charge density in real space remains a significant challenge. As shown in Theory section, ultrafast X-ray and electron diffraction techniques have the potential to directly capture the spatiotemporal evolution of a molecule’s electronic charge density. Several recent theoretical studies have investigated the use of ultrafast X-ray scattering to probe charge migration.46,75,76,77 With the ongoing development of attosecond hard XFELs worldwide,25,26 experimental realizations of attosecond X-ray diffraction are expected in the near future. Attosecond electron diffraction, although technically more challenging due to the space-charge effect, has shown promise for gas-phase implementation thanks to recent advancements in attosecond electron pulse generation and single-electron pulse techniques.28,29

As evident from Equations 5 and 9, ultrafast X-ray and electron diffraction can track charge migration, but the molecular charge density in real space (Equation 3) cannot be directly obtained from these signals as they measure the expectation values of products of charge-density operators. A recent theoretical work proposed a novel experimental scheme which combines ultrafast X-ray and electron diffraction techniques.46 This approach has the potential to overcome the aforementioned limitation and enable real-space imaging of attosecond electron dynamics in isolated molecules. This is achieved by isolating the mixed contribution, S˜mixedUED in Equation 9.

By carefully normalizing and subtracting the ultrafast X-ray (Equation 5) and electron diffraction (Equation 9) signals, one obtains Sdiff(q,t)=S˜nuclUED(q,t)+S˜mixedUED(q,t). In the charge migration regime, where nuclear motions are negligible, S˜nuclUED(q,t) is time-independent. The time-dependent difference signal is then given by ΔSdiff(q,t)=ΔS˜mixecUED(q,t)=2R[ΔσtotE(−q,t)σ0N(q)] where ΔσtotE(q,t)=σtotE(q,t)−σ0E(q) is the difference electronic charge density in q-space, and σ0E(q) is the total electronic charge density prior to the pump pulse (i.e., σtotE(q,t<0)). The time-dependent difference signal ΔSdiff(q,t) is solely contributed by the mixed nuclear-electronic term ΔS˜mixedUED. Since the molecular nuclei remain stationary during the attosecond charge migration dynamics, the resulting time-dependent difference signal forms an interference between the σtotE(q,t) and the σ0N(q). σ0N(q) serves as a static reference local oscillator. This generates a heterodyne signal without the need for an additional field, enabling direct measurement of the time-evolving electronic charge density σtotE(q,t). As the time-dependence of σtotE(q,t) in Equation 3 originates from its electronic coherence contribution (i.e., σcohE(r,t)), this provides spatial resolution of electron motions induced by purely electronic coherence. The ground-state nuclear geometry is assumed to be known a priori, obtainable through static diffraction measurements or high-level quantum chemistry calculations. Accurate determination of the ground-state nuclear geometry is crucial, as it can affect the accuracy of the reconstructed ΔσtotE(q,t). This approach allows for the inversion of the signal from momentum space to real space, generating a "molecular movie" of attosecond charge migration. This concept has been demonstrated through theoretical simulations to image the fast charge oscillation dynamics within a benzene ring of a fluorinated biphenyl molecule in real space (see Figure 3). We note that while this technique is applicable to randomly oriented samples, a high degree of alignment is desired in order to retrieve the full three-dimensional electronic charge densities rather than its one-dimensional radial distribution from an isotropic sample.Figure 3 Snapshots of the difference in electronic charge density, ΔσtotE(r,t), in real space for the 4-fluoro-4′-hydroxybiphenyl following oxygen K-edge (left) and fluorine K-edge (right) excitations

Reprinted with permission from ref. 46. Copyright 2022 by the American Chemistry Society.

One of the main challenges for implementing such an experiment is the non-trivial data analysis procedure required when subtracting two diffraction signals. Note that because X-ray diffraction is proportional to the Thomson cross section while electron diffraction is proportional to the Rutherford cross section, careful normalization is required when combining two individual measurements. This is crucial for isolating the desired mixed contribution to the signal. One possible solution is to normalize the individual diffraction signals using their intrinsic properties when q→0. It is known that the S˜XRD in Equation 5 is proportional to the square of the total number of electrons (Nel2) in the molecule when q→0, while the S˜UED vanishes when q→0 due to the opposite charges of electrons and nuclei canceling out.78 In addition, the diffraction signals should be deconvoluted with their respective instrument functions in the subtraction procedure to avoid errors coming from pulse-length deviations.

Probing vibronic coherences at conical intersections

Conical intersections (CIs) are important regions where two or more adiabatic potential energy surfaces become degenerate, resulting in strong non-adiabatic couplings between them. Because these regions allow for efficient, non-radiative electronic relaxations, they play critical roles in virtually all photochemical and photophysical processes. Conventional experimental signatures of CIs are primarily based on the transfer of electronic populations between electronic states. Directly observing the passage through CIs via transient vibronic coherences remains challenging, as many contemporary ultrafast measurements are dominated by much stronger signals from electronic populations that are not specific to CIs.

As manifested in Equation 8, time-resolved X-ray diffraction contains a mixed elastic-inelastic scattering term S˜cohXRD(q,t) originating from vibronic coherences. However, this mixed coherence term is found to be significantly weaker than the dominating contributions from electronic populations S˜popXRD(q,t) during CI passage, making its experimental realization very challenging. A recent theoretical article has shown that this major obstacle in standard ultrafast X-ray diffraction can be overcome by measuring the rotationally averaged time-resolved X-ray diffraction using twisted X-ray beams carrying orbital angular momentum (OAM).79 Twisted beams, also known as vortex or OAM beams, possess a helical spatial wavefront that twists along the beam propagation direction, as shown in Figure 4A, independently of the beam polarization state. Various strategies have been developed to generate intense, hard X-ray twisted beams,80,81,82 enabling ultrafast spectroscopic measurements of molecules using these OAM beams.83 Twisted X-ray scattering provides structural information in addition to energy observables in molecules, showing promise in revealing both the spatial and temporal profiles of transient vibronic coherences generated at CIs.Figure 4 Illustrations and simulations of twisted X-ray scattering

(A) Helical spatial wavefronts of twisted beams carrying various OAMs (L = +1 and L = −1).

(B) Reaction pathway of thiophenol S-H photodissociation. Two CIs are marked with open circles.

(C) The difference of simulated rotationally averaged diffraction signals, ΔSl(q,T), of thiophenol at 5.3 fs delay time for population and coherence contribution, respectively.

(D) Comparison of the simulated signal, ΔSl(q,T), integrated over q and the total coherence magnitude of the vibronic coherences at two CIs of thiophenol photodissociation calculated by quantum dynamics. Adapted with permission from ref. 79. Copyright 2022 by the American Physical Society.

The vector potential AX of OAM beams is given by(Equation 10) AX(r,t)=E(t)Al(r,z,ϕ)=E(t)A(r,z)eilϕ

where A(r,z) is the radial profile of the beam at height z and E(t) is the temporal profile. Over the molecular scale, the radial profile A(r,z) can be assumed as a constant. The angular index l is called the topological charge of the twisted beam. Twisted beams are eigenstates of the angular momentum operator, and possess an orbital angular momentum (OAM) of lℏ per photon. Here the beam profile is expressed in cylindrical coordinates, where r is the radial distance, ϕ is angular coordinate and z is axial coordinate. The OAM beam polarization can be kept linear in the proposed setup and gets absorbed in a Lorentz-polarization factor |ϵx·ϵs|2 in Equation 4, where ϵx and ϵs are polarization vectors of the X-ray probe pulse and scattered photon.

The time-resolved twisted X-ray diffraction signal is obtained by substituting Equation 10 in Equations 4 and 5,79(Equation 11) Sl(q,T)∝∫dt|E(t−T)|2Sl˜(q,t)

where Sl˜(q,t)=〈σˆl†(q,t)σˆl(q,t)〉 and σˆl(q)=∫σˆE(r)eilϕeiq·rdr is the momentum-space electronic charge-density operator carrying OAM. Similar to the Equations 6, 7, and 8, the time-resolved twisted X-ray diffraction signal can be partitioned into the sum of contributions from electronic populations and coherences,(Equation 12) S˜l(q,t)=S˜lpop(q,t)+S˜lcoh(q,t)

The full derivation of the signal has been discussed previously.79 It has been shown that when taking the difference of the rotationally averaged diffraction signal measured with the positive and negative OAM beams, i.e., ΔSl(q,T)=〈Sl(q,T)〉Ω−〈S−l(q,T)〉Ω, the contributions of electronic populations, S˜lpop(q,t), cancel out and only the desired coherence signal, S˜lcoh(q,t), survive. Here q is the norm of the momentum transfer vector q and 〈…〉Ω denotes the rotationally averaged diffraction signal.

Previous simulation has demonstrated the concept by applying it to thiophenol photodissociation shown in Figure 4B involving two CIs (S2/S1 and S1/S0 CI) calculated by exact quantum dynamical simulations.79 The simulated quantum dynamics was used to simulate the time-resolved rotationally averaged diffraction signals, Sl(q,T) for l=1 and l=−1 OAM X-ray beams. Figure 4C shows their difference signal at a chosen delay time which demonstrates that the population contribution to the isotropic difference signal ΔSlpop(q,T) vanishes so that only coherence signal contributes to the total difference signal ΔSl(q,T). The time-resolved difference signal, ΔSl(q,T), was further simulated for the entire thiophenol photodissociation dynamics. Figure 4D shows the time-dependent absolute difference signal integrated over q. For comparison, the sum of the S2/S1 and S1/S0 vibronic coherence magnitudes given by the quantum dynamical simulations is also shown. The integrated difference signal clearly resembles the time-dependent vibronic coherences in the molecule. By implementing this difference measurement scheme, one could potentially directly monitor the passage through CIs imprinted in the transient vibronic coherences.

A major challenge is the inevitably weak magnitude of the coherence signal. The signal shown in Figure 4C is about three orders of magnitude smaller than the detection limit reported so far in existing ultrafast gas-phase X-ray diffraction experiments.14 These experiments were performed at XFEL facilities with a repetition rate of 120 Hz. To achieve a high signal-to-noise ratio for resolving the desired electronic coherence signal, the ongoing development of high-repetition-rate (MHz) hard X-ray sources is beneficial for the realization of the proposed experiment. Alternatively, employing a weak resonant infrared field to enhance coherence signatures in diffraction signals could be helpful.84

It is worth pointing out that since the proposed technique essentially relies solely on the coherence contribution to the diffraction signal (i.e., S˜cohXRD in Equation 8), its application is more general, going beyond vibronic coherences at CIs. It can be applied to spatially resolve many other fundamental quantum coherence phenomena in molecules, including attosecond charge migration discussed in probing attosecond electron dynamics driven by electronic coherences.

Conclusion

Ultrafast gas-phase X-ray scattering and electron diffraction have proven to be powerful tools for investigating the nuclear dynamics of molecules with atomic spatial and temporal resolution. Recent theoretical developments, including the use of twisted X-ray beams and the combination of X-ray and electron diffraction techniques, unveiled the potential to extend these methods into the attosecond domain. This would allow for the direct observation of electronic and vibronic coherences that play critical roles in photochemical processes. By resolving charge migration in real-space and time, and disentangling the intricate coupling between electronic and nuclear degrees of freedom at conical intersections, these emerging techniques will provide valuable insights into the fundamental mechanisms underlying chemical reactivity. As attosecond X-ray and electron sources continue to advance, we expect these tools to open new avenues for exploring the quantum dynamics of molecules at their most fundamental level.

To fully harness the potential of ultrafast gas-phase diffraction, continued advancements in instrumentation, theoretical modeling, and data analysis are necessary. Overcoming the challenges associated with generating intense and stable attosecond pulses, improving detector performance, and developing robust data inversion algorithms will require a concerted effort from the scientific community. By addressing these limitations, we anticipate that ultrafast gas-phase diffraction will provide unprecedented insights into the quantum dynamics of molecules, ultimately paving the way for the rational design and control of chemical reactions at the attosecond scale. This will not only deepen our understanding of the fundamental principles governing chemical reactivity but also have far-reaching implications for fields such as photocatalysis, energy conversion, and quantum technologies.

Limitations of the study

This article reflects the authors' perspective on the current state and future potential of ultrafast gas-phase X-ray scattering and electron diffraction techniques. It is important to acknowledge that challenges remain in the development and application of these methods. Generating sufficiently intense and stable attosecond pulses, improving detector performance, and developing robust data inversion algorithms will require a concerted effort from the scientific community. Addressing these limitations will be crucial for realizing the full potential of ultrafast gas-phase diffraction in unraveling the quantum dynamics of molecules at the attosecond scale.

Acknowledgments

This research was supported in part by 10.13039/100000888 W. M. Keck Foundation through computing resources at the W. M. Keck Laboratory for Integrated Biology at UC San Diego.

Author contributions

Conceptualization, Z.T. and H.Y.; investigation, Z.T., R.J. and H.Y.; writing – original draft, Z.T. and H.Y.; writing – review and editing, Z.T., R.J. and H.Y.; and supervision, H.Y.

Declaration of interests

The authors declare no competing interests.
==== Refs
References

1 Tannor D.J. Kosloff R. Rice S.A. Coherent pulse sequence induced control of selectivity of reactions: Exact quantum mechanical calculations J. Chem. Phys. 85 1986 5805 5820
2 Zewail A.H. Femtochemistry: Atomic-scale dynamics of the chemical bond J. Phys. Chem. A 104 2000 5660 5694
3 Scholes G.D. Fleming G.R. Chen L.X. Aspuru-Guzik A. Buchleitner A. Coker D.F. Engel G.S. van Grondelle R. Ishizaki A. Jonas D.M. Using coherence to enhance function in chemical and biophysical systems Nature 543 2017 647 656 28358065
4 Cao J. Cogdell R.J. Coker D.F. Duan H.G. Hauer J. Kleinekathöfer U. Jansen T.L.C. Mančal T. Miller R.J.D. Ogilvie J.P. Quantum biology revisited Sci. Adv. 6 2020 eaaz4888
5 Wasielewski M.R. Forbes M.D.E. Frank N.L. Kowalski K. Scholes G.D. Yuen-Zhou J. Baldo M.A. Freedman D.E. Goldsmith R.H. Goodson T. 3rd Exploiting chemistry and molecular systems for quantum information science Nat. Rev. Chem 4 2020 490 504 37127960
6 Williamson J.C. Cao J. Ihee H. Frey H. Zewail A.H. Clocking transient chemical changes by ultrafast electron diffraction Nature 386 1997 159 162
7 King W.E. Campbell G.H. Frank A. Reed B. Schmerge J.F. Siwick B.J. Stuart B.C. Weber P.M. Ultrafast Electron Microscopy in Materials Science, Biology, and Chemistry J. Appl. Phys. 36 2005 111101 111127
8 Gaffney K.J. Chapman H.N. Imaging atomic structure and dynamics with ultrafast x-ray scattering Science 316 2007 1444 1448 17556577
9 Sciaini G. Miller R.J.D. Femtosecond electron diffraction: heralding the era of atomically resolved dynamics Rep. Prog. Phys. 74 2011 096101
10 Ischenko A.A. Weber P.M. Miller R.J.D. Capturing Chemistry in Action with Electrons: Realization of Atomically Resolved Reaction Dynamics Chem. Rev. 117 2017 11066 11124 28590727
11 Ihee H. Lobastov V.A. Gomez U.M. Goodson B.M. Srinivasan R. Ruan C.Y. Zewail A.H. Direct Imaging of Transient Molecular Structures with Ultrafast Diffraction Science 291 2001 458 462 11161194
12 Ruan C.Y. Lobastov V.A. Srinivasan R. Goodson B.M. Ihee H. Zewail A.H. Ultrafast diffraction and structural dynamics: The nature of complex molecules far from equilibrium Proc. Natl. Acad. Sci. USA 98 2001 7117 7122 11404473
13 Dudek R.C. Weber P.M. Ultrafast Diffraction Imaging of the Electrocyclic Ring-Opening J. Phys. Chem. 105 2001 4167 4171
14 Yong H. Kirrander A. Weber P.M. Time-resolved x-ray scattering of excited state structure and dynamics in Structural Dynamics with X-ray and Electron Scattering Royal Society of Chemistry 25 2023 344 373
15 Centurion M. Wolf T.J.A. Yang J. Ultrafast imaging of molecules with electron diffraction Annu. Rev. Phys. Chem. 73 2022 21 42 34724395
16 Odate A. Kirrander A. Weber P.M. Minitti M.P. Brighter, faster, stronger: ultrafast scattering of free molecules Adv. Phys. X 8 2023 2126796
17 Lee Y. Oang K.Y. Kim D. Ihee H. A comparative review of time-resolved x-ray and electron scattering to probe structural dynamics Struct. Dyn. 11 2024 031301
18 Wang Q. Yun L. Yang J. Ultrafast Molecular Movies: Probing Chemical Dynamics with Femtosecond Electron and X-Ray Diffraction CCS Chem. 6 2024 1092 1109
19 Goulielmakis E. Loh Z.H. Wirth A. Santra R. Rohringer N. Yakovlev V.S. Zherebtsov S. Pfeifer T. Azzeer A.M. Kling M.F. Real-time observation of valence electron motion Nature 466 2010 739 743 20686571
20 Calegari F. Ayuso D. Trabattoni A. Belshaw L. De Camillis S. Anumula S. Frassetto F. Poletto L. Palacios A. Decleva P. Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses Science 346 2014 336 339 25324385
21 Kraus P.M. Mignolet B. Baykusheva D. Rupenyan A. Horný L. Penka E.F. Grassi G. Tolstikhin O.I. Schneider J. Jensen F. Measurement and laser control of attosecond charge migration in ionized iodoacetylene Science 350 2015 790 795 26494175
22 Månsson E.P. Latini S. Covito F. Wanie V. Galli M. Perfetto E. Stefanucci G. Hübener H. De Giovannini U. Castrovilli M.C. Real-time observation of a correlation-driven sub 3 fs charge migration in ionized adenine Commun. Chem. 4 2021 73 36697766
23 Barillot T. Alexander O. Cooper B. Driver T. Garratt D. Li S. Al Haddad A. Sanchez-Gonzalez A. Agåker M. Arrell C. Correlation-driven transient hole dynamics resolved in space and time in the isopropanol molecule Phys. Rev. X 11 2021 031048
24 Li S. Driver T. Rosenberger P. Champenois E.G. Duris J. Al-Haddad A. Averbukh V. Barnard J.C.T. Berrah N. Bostedt C. Attosecond coherent electron motion in Auger-Meitner decay Science 375 2022 285 290 34990213
25 Duris J. Li S. Driver T. Champenois E.G. MacArthur J.P. Lutman A.A. Zhang Z. Rosenberger P. Aldrich J.W. Coffee R. Tunable isolated attosecond x-ray pulses with gigawatt peak power from a free-electron laser Nat. Photonics 14 2020 30 36
26 Xiao Y. Feng C. Liu B. Generating isolated attosecond x-ray pulses by wavefront control in a seeded free-electron laser Ultrafast Sci. 2022 2022 9812478
27 Franz P. Li S. Driver T. Robles R. Cesar D. Isele E. Guo Z. Wang J. Duris J. Larsen K. Terawatt-scale attosecond X-ray pulses from a cascaded superradiant free-electron laser Nat. Photon 18 2024 698 703
28 Baum P. On the physics of ultrashort single-electron pulses for time-resolved microscopy and diffraction Chem. Phys. 423 2013 55 61
29 Morimoto Y. Baum P. Diffraction and microscopy with attosecond electron pulse trains Nat. Phys. 14 2018 252 256
30 Merritt I.C.D. Jacquemin D. Vacher M. Attochemistry: Is controlling electrons the future of photochemistry? J. Phys. Chem. Lett. 12 2021 8404 8415 34436903
31 Calegari F. Martin F. Open questions in attochemistry Commun. Chem. 6 2023 184 37666969
32 Scholes G.D. Fleming G.R. Olaya-Castro A. van Grondelle R. Lessons from nature about solar light harvesting Nat. Chem. 3 2011 763 774 21941248
33 Brieke C. Rohrbach F. Gottschalk A. Mayer G. Heckel A. Light-Controlled Tools Angew. Chem. Int. Ed. 51 2012 8446 8476
34 Szymański W. Beierle J.M. Kistemaker H.A.V. Velema W.A. Feringa B.L. Reversible photocontrol of biological systems by the incorporation of molecular photoswitches Chem. Rev. 113 2013 6114 6178 23614556
35 Ben-Nun M. Martínez T.J. Weber P.M. Wilson K.R. Direct Imaging of Excited Electronic States Using Diffraction Techniques: Theoretical Considerations Chem. Phys. Lett. 263 1996 405 414
36 Ben-Nun M. Cao J. Wilson K.R. Ultrafast X-Ray and Electron Diffraction: Theoretical Considerations J. Phys. Chem. A 101 1997 8743 8761
37 Cao J. Wilson K.R. Ultrafast X-Ray Diffraction Theory J. Phys. Chem. A 102 1998 9523 9530
38 Henriksen N.E. Møller K.B. On the Theory of Time-Resolved X-Ray Diffraction J. Phys. Chem. B 112 2008 558 567 18052363
39 Dixit G. Vendrell O. Santra R. Imaging Electronic Quantum Motion with Light Proc. Natl. Acad. Sci. USA 109 2012 11636 11640 22753505
40 Bennett K. Kowalewski M. Rouxel J.R. Mukamel S. Monitoring Molecular Nonadiabatic Dynamics with Femtosecond X-Ray Diffraction Proc. Natl. Acad. Sci. USA 115 2018 6538 6547 29891703
41 Simmermacher M. Henriksen N.E. Møller K.B. Moreno Carrascosa A. Kirrander A. Electronic Coherence in Ultrafast X-Ray Scattering from Molecular Wave Packets Phys. Rev. Lett. 122 2019 073003
42 Simmermacher M. Moreno Carrascosa A. E Henriksen N. B Møller K. Kirrander A. Theory of Ultrafast X-Ray Scattering by Molecules in the Gas Phase J. Chem. Phys. 151 2019 174302
43 Rouxel J.R. Keefer D. Mukamel S. Signatures of electronic and nuclear coherences in ultrafast molecular x-ray and electron diffraction Struct. Dyn. 8 2021 014101
44 Yong H. Keefer D. Mukamel S. Novel Ultrafast Molecular Imaging Based on the Combination of X-ray and Electron Diffraction J. Phys. Chem. A 127 2023 835 841 36650121
45 Dixit G. Slowik J.M. Santra R. Theory of time-resolved nonresonant x-ray scattering for imaging ultrafast coherent electron motion Phys. Rev. 89 2014 043409
46 Yong H. Sun S. Gu B. Mukamel S. Attosecond charge migration in molecules imaged by combined x-ray and electron diffraction J. Am. Chem. Soc. 144 2022 20710 20716 36318702
47 Brockway L.O. Electron Diffraction by Gas Molecules Rev. Mod. Phys. 8 1936 231 266
48 Emma P. Akre R. Arthur J. Bionta R. Bostedt C. Bozek J. Brachmann A. Bucksbaum P. Coffee R. Decker F.J. First lasing and operation of an ångstrom-wavelength free-electron laser Nat. Photonics 4 2010 641 647
49 Liu S. Decking W. Kocharyan V. Saldin E. Serkez S. Shayduk R. Sinn H. Geloni G. Preparing for high-repetition rate hard x-ray self-seeding at the European X-ray Free Electron Laser: Challenges and opportunities Phys. Rev. Accel. Beams 22 2019 060704
50 Küpper J. Stern S. Holmegaard L. Filsinger F. Rouzée A. Rudenko A. Johnsson P. Martin A.V. Adolph M. Aquila A. X-ray diffraction from isolated and strongly aligned gas-phase molecules with a free-electron laser Phys. Rev. Lett. 112 2014 083002
51 Minitti M.P. Budarz J.M. Kirrander A. Robinson J. Lane T.J. Ratner D. Saita K. Northey T. Stankus B. Cofer-Shabica V. Toward structural femtosecond chemical dynamics: imaging chemistry in space and time Faraday Discuss 171 2014 81 91 25415842
52 Minitti M.P. Budarz J.M. Kirrander A. Robinson J.S. Ratner D. Lane T.J. Zhu D. Glownia J.M. Kozina M. Lemke H.T. Imaging molecular motion: femtosecond x-ray scattering of an electrocyclic chemical reaction Phys. Rev. Lett. 114 2015 255501
53 Budarz J.M. Minitti M.P. Cofer-Shabica D.V. Stankus B. Kirrander A. Hastings J.B. Weber P.M. Observation of femtosecond molecular dynamics via pump–probe gas phase x-ray scattering J. Phys. B Atom. Mol. Opt. Phys. 49 2016 034001
54 Stankus B. Yong H. Ruddock J. Ma L. Carrascosa A.M. Goff N. Boutet S. Xu X. Zotev N. Kirrander A. Advances in ultrafast gas-phase x-ray scattering J. Phys. B Atom. Mol. Opt. Phys. 53 2020 234004
55 Stankus B. Budarz J.M. Kirrander A. Rogers D. Robinson J. Lane T.J. Ratner D. Hastings J. Minitti M.P. Weber P.M. Femtosecond photodissociation dynamics of 1,4-diiodobenzene by gas-phase X-ray scattering and photoelectron spectroscopy Faraday Discuss 194 2016 525 536 27711844
56 Glownia J.M. Natan A. Cryan J.P. Hartsock R. Kozina M. Minitti M.P. Nelson S. Robinson J. Sato T. van Driel T. Self-Referenced Coherent Diffraction X-Ray Movie of Ångstrom- and Femtosecond-Scale Atomic Motion Phys. Rev. Lett. 117 2016 153003
57 Stankus B. Yong H. Zotev N. Ruddock J.M. Bellshaw D. Lane T.J. Liang M. Boutet S. Carbajo S. Robinson J.S. Ultrafast X-ray scattering reveals vibrational coherence following Rydberg excitation Nat. Chem. 11 2019 716 721 31285542
58 Ware M.R. Glownia J.M. Al-Sayyad N. O’Neal J.T. Bucksbaum P.H. Characterizing dissociative motion in time-resolved x-ray scattering from gas-phase diatomic molecules Phys. Rev. 100 2019 033413
59 Yong H. Moreno Carrascosa A. Ma L. Stankus B. Minitti M.P. Kirrander A. Weber P.M. Determination of excited state molecular structures from time-resolved gas-phase X-ray scattering Faraday Discuss 228 2021 104 122 33595043
60 Yong H. Xu X. Ruddock J.M. Stankus B. Carrascosa A.M. Zotev N. Bellshaw D. Du W. Goff N. Chang Y. Ultrafast X-ray scattering offers a structural view of excited-state charge transfer Proc. Natl. Acad. Sci. USA 118 2021 e2021714118
61 Yong H. Zotev N. Ruddock J.M. Stankus B. Simmermacher M. Carrascosa A.M. Du W. Goff N. Chang Y. Bellshaw D. Observation of the molecular response to light upon photoexcitation Nat. Commun. 11 2020 2157 32358535
62 Ruddock J.M. Zotev N. Stankus B. Yong H. Bellshaw D. Boutet S. Lane T.J. Liang M. Carbajo S. Du W. Simplicity Beneath Complexity: Counting Molecular Electrons Reveals Transients and Kinetics of Photodissociation Reactions Angew. Chem. Int. Ed. 58 2019 6371 6375
63 Ruddock J.M. Yong H. Stankus B. Du W. Goff N. Chang Y. Odate A. Carrascosa A.M. Bellshaw D. Zotev N. A deep UV trigger for ground-state ring-opening dynamics of 1,3-cyclohexadiene Sci. Adv. 5 2019 eaax6625
64 Yong H. Ruddock J.M. Stankus B. Ma L. Du W. Goff N. Chang Y. Zotev N. Bellshaw D. Boutet S. Scattering off molecules far from equilibrium J. Chem. Phys. 151 2019 084301
65 Yang J. Guehr M. Shen X. Li R. Vecchione T. Coffee R. Corbett J. Fry A. Hartmann N. Hast C. Diffractive Imaging of Coherent Nuclear Motion in Isolated Molecules Phys. Rev. Lett. 117 2016 153002
66 Yang J. Zhu X. Wolf T.J.A. Li Z. Nunes J.P.F. Coffee R. Cryan J.P. Gühr M. Hegazy K. Heinz T.F. Imaging CF3I Conical Intersection and Photodissociation Dynamics with Ultrafast Electron Diffraction Science 361 2018 64 67 29976821
67 Wolf T.J.A. Sanchez D.M. Yang J. Parrish R.M. Nunes J.P.F. Centurion M. Coffee R. Cryan J.P. Gühr M. Hegazy K. The Photochemical Ring-Opening of 1,3-Cyclohexadiene Imaged by Ultrafast Electron Diffraction Nat. Chem. 11 2019 504 509 30988415
68 Liu Y. Sanchez D.M. Ware M.R. Champenois E.G. Yang J. Nunes J.P.F. Attar A. Centurion M. Cryan J.P. Forbes R. Rehybridization Dynamics into the Pericyclic Minimum of an Electrocyclic Reaction Imaged in Real-Time Nat. Commun. 14 2023 2795 37202402
69 Razmus W.O. Acheson K. Bucksbaum P. Centurion M. Champenois E. Gabalski I. Hoffman M.C. Howard A. Lin M.F. Liu Y. Multichannel Photodissociation Dynamics in CS2 Studied by Ultrafast Electron Diffraction Phys. Chem. Chem. Phys. 24 2022 15416 15427 35707953
70 Yang J. Zhu X. F Nunes J.P. Yu J.K. Parrish R.M. Wolf T.J.A. Centurion M. Gühr M. Li R. Liu Y. Simultaneous Observation of Nuclear and Electronic Dynamics by Ultrafast Electron Diffraction Science 368 2020 885 889 32439793
71 Champenois E.G. List N.H. Ware M. Britton M. Bucksbaum P.H. Cheng X. Centurion M. Cryan J.P. Forbes R. Gabalski I. Femtosecond Electronic and Hydrogen Structural Dynamics in Ammonia Imaged with Ultrafast Electron Diffraction Phys. Rev. Lett. 131 2023 143001
72 Kling M.F. Vrakking M.J.J. Attosecond electron dynamics Annu. Rev. Phys. Chem. 59 2008 463 492 18031218
73 Nisoli M. Decleva P. Calegari F. Palacios A. Martín F. Attosecond electron dynamics in molecules Chem. Rev. 117 2017 10760 10825 28488433
74 Wörner H.J. Arrell C.A. Banerji N. Cannizzo A. Chergui M. Das A.K. Hamm P. Keller U. Kraus P.M. Liberatore E. Charge migration and charge transfer in molecular systems Struct. Dyn. 4 2017 061508
75 Yong H. Cavaletto S.M. Mukamel S. Ultrafast valence-electron dynamics in oxazole monitored by x-ray diffraction following a stimulated x-ray Raman excitation J. Phys. Chem. Lett. 12 2021 9800 9806 34606289
76 Moreno Carrascosa A. Yang M. Yong H. Ma L. Kirrander A. Weber P.M. Lopata K. Mapping static core-holes and ring-currents with x-ray scattering Faraday Discuss 228 2021 60 81 33605956
77 Giri S. Tremblay J.C. Dixit G. Imaging charge migration in chiral molecules using time-resolved x-ray diffraction Phys. Rev. 104 2021 053115
78 Moreno Carrascosa A. Yong H. Crittenden D.L. Weber P.M. Kirrander A. Ab initio calculation of total x-ray scattering from molecules J. Chem. Theor. Comput. 15 2019 2836 2846
79 Yong H. Rouxel J.R. Keefer D. Mukamel S. Direct monitoring of conical intersection passage via electronic coherences in twisted x-ray diffraction Phys. Rev. Lett. 129 2022 103001
80 Hemsing E. Marinelli A. Rosenzweig J.B. Generating optical orbital angular momentum in a high-gain free-electron laser at the first harmonic Phys. Rev. Lett. 106 2011 164803
81 Bahrdt J. Holldack K. Kuske P. Müller R. Scheer M. Schmid P. First observation of photons carrying orbital angular momentum in undulator radiation Phys. Rev. Lett. 111 2013 034801
82 Seiboth F. Kahnt M. Lyubomirskiy M. Seyrich M. Wittwer F. Ullsperger T. Nolte S. Batey D. Rau C. Schroer C.G. Refractive hard x-ray vortex phase plates Opt. Lett. 44 2019 4622 4625 31517948
83 Rouxel J.R. Rösner B. Karpov D. Bacellar C. Mancini G.F. Zinna F. Kinschel D. Cannelli O. Oppermann M. Svetina C. Hard x-ray helical dichroism of disordered molecular media Nat. Photonics 16 2022 570 574
84 Keefer D. Rouxel J.R. Aleotti F. Segatta F. Garavelli M. Mukamel S. Diffractive imaging of conical intersections amplified by resonant infrared fields J. Am. Chem. Soc. 143 2021 13806 13815 34402612
